#188251

order	512
nilpotency class	3
isoclinism	isoclinic class of #24468
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#379
prime radical	#62810	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2230 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

(2,7)(4,5)(8,15)(9,12)(10,13)(14,16)(17,18,20,24,21,23,27,30)(19,22,26,28,25,29,31,32);(1,2)(3,8)(4,7)(5,12)(6,9)(10,14)(11,16)(13,15);(1,3)(2,8)(4,10)(5,13)(6,11)(7,14)(9,16)(12,15)(17,19)(18,22)(20,26)(21,25)(23,29)(24,28)(27,31)(30,32);(1,4,6,5)(2,7,9,12)(3,10,11,13)(8,14,16,15)(18,23)(19,25)(24,30)(26,31)

Polycyclic presentation:

2^9|a^2=g,d^2=h,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=dhi,e^a=eh,e^b=eh,f^b=fh,d^c=di,e^c=eh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 95447);